Let $\theta$ be a real number with $0 < \theta < \pi$. Let $A$ and $O$ be distinct points in three-dimensional space. Let $S$ be the set of all points $P$ in space with both $OP < 1$ and $\angle AOP <\theta$. Find the volume of $S$ in terms of $\theta$.

(This problem is possible to solve with calculus, but there’s also a very nice solution that avoids it.)